When one numerically crosses momenta in a given helicity amplitude (for example, in order to derive a different helicity from a known one) the above procedure needs to be slightly modified. Since in this case one cannot rely on the set of pre-crossed UTMs anymore, one has to treat correctly the signs of both the parity-odd variable \(\varepsilon _5\)  and the parity-odd functions within the transcendental basis \(t_e\) . Specifically, if an odd permutation is involved the signs of both flip. As we previously remarked, the transcendental functions of ref. [1]} allow for the numerical evaluation of crossings only if the momenta are not switched between the initial and the final states.
